@YassineGuerboussa thank you for your answers. About the Frattini subgroup, the answer is very easy, i should not ask for. Instead i've no answers about $\Omega_{1}(Z_{2}(G))$.

I'm grateful for this answer.

@Derek Holt I'm really sorry for my unfocused question. I'm analyzing the elements of order p of the automorphism group of a group of fixed small coclass, it remains to me to work on a finite p-group that is UL-equivalent (otherwise i have just solved the problem), so i was searching existing results on this kind of group, and i asked since i'd been not able to find them. Thank you for your attention.

@Nick Gill I'm really sorry for my unfocused question. I'm analyzing the elements of order $p$ of the automorphism group of a group of fixed small coclass, it remains to me to work on a finite $p$-group that is UL-equivalent (otherwise i have just solved the problem), so i was searching existing results on this kind of group, and i asked since i'd been not able to find them. Thank you for your attention.

@KhalidBou-Rabee Thank you for the comment. I've seen yesterday the thread that you linked to me, there are listed in it examples of groups which have the UL-equivalence, unfortunately i was searching results that assume the UL-equivalence like an hypothesis.

@NickGill Thank you for the comment. I've seen on group props before write the question, unfortunately i'd not able to find anything useful since the only weaker property listed is the nilpotence, but i'm studying a problem on a $p$- group, and all $p$-groups are nilpotents.

@AlirezaAbdollahi thank you for your answer.

@DerekHolt i'm grateful for your answer, i had not found this results in my books.

@ArturoMagidin Ok, i'm grateful for your answers. I'm sorry for my mistakes writing the question, it will never happen again. Thank you again, Marco.

Thank you for the answer. I agree that when $[G,G] \subseteq Z(G)$ we have that $C_{G}(G') =G$, but in class greater than $p$ ($G$ not regular) this is not necessarly true. I'm wrong?